Fundamentals of Logic
Statements/Propositions
– Sentences that are true or false but
not both. (Just like a simple boolean expression or conditional
expression in a programming language.)
For our purposes, statements will simply be denoted by
lowercase letters of the alphabet, typically p, q, and r. I will
refer to these as boolean variables as well.
Given simple statements, we can construct more complex
statements using logical connectives. Here are 4 logical
connectives we will use:
1)
Conjunction: This is denoted by the ‘
∧
’ symbol. The
statement p
∧
q is read as “p and q.” Only if both the values
of p and q are true does this expression evaluate to true.
Otherwise it is false.
2)
Disjunction: This is denoted by the ‘
∨
’ symbol. The p
∨
q is
read as “p or q.” As long as at least one of the values of p or
q is true, the entire expression is true
3)
Implication: This is denoted by the ‘
⇒
’ symbol. The
statement p
⇒
q is read as “p implies q”. Essentially, in a
programming language, this logic is captured in an if then
statement. If p is true, the q must be true. However, if p is
not true, there is no guarantee of the truth of q. An
important observation to note: when statements are
combined with an implication, there is no need for there to
be a causal relationship between the two for the implication
to be true.
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Consider the following implications:
If my bread is green, then I will not eat it.
Here, if the bread is not green, that does not guarantee that I
will eat it. Perhaps it is wheat bread and I hate wheat bread.
All I know is that if my bread is green, I will definitely NOT
eat it.
If Pluto is the largest planet in our solar system, then pigs will
fly out of my butt.
Wayne would actually be making a correct implication here,
(assuming that we currently have accurate knowledge about
our solar system...) Since the first part of our implication is
false, the entire implication is automatically true.
4)
Biconditional: This is denoted by the ‘
⇔
’ symbol, and the
statement p
⇔
q is read “p if and only if q.” The phrase “if
and only if” is often abbreviated as “iff”. Breaking this
down into pieces, this means that p
⇒
q AND q
⇐
p. Hence
exactly when p is true, q is true. (And in all cases where p is
false, q must be false as well.)
Here is an example of a biconditional:
If and only if my alarm clock rings in the morning, then I will
attend my morning classes.
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 Spring '09
 Logic, Daunte Culpepper

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